ar X iv : h ep - t h / 99 08 19 5 v 1 2 9 A ug 1 99 9 Gauge Field Theories on a ⊥ lattice

In these notes, the transverse (⊥) lattice approach is presented as a means to control the k → 0 divergences in light-front QCD. Technical difficulties of both the canonical compact formulation as well as the non-compact formulation of the ⊥ lattice motivate the color-dielectric formulation, where the link fields are linearized. INTRODUCTION The main subject of these notes are difficulties associated with the formulation of gauge field theories on a transverse (⊥) lattice using light-front (LF) quantization. Because of these difficulties, the reader may wonder about the advantages of this approach — particularly given the successes of Euclidean lattice gauge theory (LGT). The primary motivations for formulating QCD in this framework is that LF quantization is the most physical approach towards a microscopic description of the parton distributions measured in deep-inelastic scattering as well as many other hard processes. 1 It is important to emphasize this fact in this brief introduction since it explains why LF quantization of QCD and the ⊥ lattice should be investigated as a possible alternative to Euclidean and Hamiltonian LGT formulations — despite the difficulties that will be discussed in the remainder of these notes. Why LF gauge? Although the choice of quantization hyperplane and the choice of gauge are in principle independent issues, the so-called LF gauge (A = 0) turns out to be highly preferable for the canonical formulation of LFQCD. The main reason is that in the kinetic energy term for ~ A⊥ (from − 4F Fμν) Lkin,A⊥ = D+ ~ A⊥D− ~ A⊥ = (∂+ − igA+) ~ A⊥ (∂− − igA−) ~ A⊥, (1) the term multiplying the ‘time’ derivative of ~ A⊥ (i.e. ∂+ ~ A⊥) contains also A− = A . Therefore, the canonical momenta 1) This and other motivations are discussed in more detail in Ref. [1] and in references therein. Π = ∂L ∂(∂+A⊥) = (∂− − igA−)A⊥, (2) which are the LF analog to Π = ∂L ∂(∂0A⊥) in equal time quantization, are “simple” (i.e. linear in the fields) if and only if A− = A + = 0. Therefore, in order to avoid having to deal with a system that has to satisfy nonlinear constraints 2 one normally selects A = 0 gauge before quantizing in LF coordinates. However, this choice of gauge is not entirely free of problems. To illustrate this fact, let us start from the Euler-Lagrange equation for A− in QED 3 −∂ −A = gJ + ∂−~ ∂⊥ ~ A⊥ ≡ gJ̃ (3) (the LF analog to the Poisson equation), which is also a constraint equation. It is convenient to eliminate A−, using the solution to Eq.(3), i.e. A−(x−, ~x⊥) = − 2 ∞ −∞ dy − |x− − y−| J̃+(y−, ~x⊥), yielding an instantaneous interaction term V inst = − 2 4 ∫ ∞ −∞ dx− ∫ ∞ −∞ dy− ∫